Transfer functions (input-output reactions) are the basic and recognized parameters which describe any system, including radioelectronic, electronic, mechanical, optical systems (and the like), their combinations, and derivative systems.
Two standard subclasses of these functions are referred to as amplitude and phase transfer functions; these correspondingly describe the amplitude to amplitude, or the phase to phase behavior of the system under analysis. Being frequency dependent, both of the named functions form corresponding subclasses of the so-called "spectral" characteristics of the systems. Usually, given the set of the above mentioned functions, the system becomes fully described.
But the task of evaluation of transfer functions represents a very sophisticated problem both mathematically and experimentally for all of the systems mentioned above.
Without describing the numerous methods for evaluation of the said characteristics and without itemizing their disadvantages, some observations should be considered. The evaluation of the basic functions which describe a system represents such a sophisticated problem, that usually the system is "linearized", i.e., a linear approximation of the transfer function(s) is applied. Then the system is considered to be linear.
It is recognized though, that non-linear systems provide the most effective functioning results. Linearization proves additionally, to be absolutely ineffective in the evaluation of phase transfer functions; phase-systems are always non-linear, even when their transfer functions are linearized. That disadvantage is easily seen when phase-registering media (or phase-transceiving networks) are analyzed.
For phase media (such as optical video disks and audio-disks, optical recording media for computers, media for standard audio recording, etc.) the said problem manifests itself in the fact that the corresponding functions are evaluated experimentally, using the "trial and error method". Stated simply, many experiments must be performed, during which separate and finite inputs are applied, while the corresponding outputs are measured.
Such evaluations are always time-consuming, complicated experimentally, and inaccurate. This is especially true in optical networks where relief media shifts of about of 1/4 of a wavelength should be accounted for. Even with such problems being present, the basic functions of the system(s) are evaluated which provide the basis for future qualitative and quantitative assumptions and predictions.
The basic reason affecting the mentioned complexity is that the general mathematical description of complex (amplitude-phase) non-linear systems is very sophisticated. Though the basic equations can be written down, they represent in the general case, a set of integral equations of N-th order, where additionally, the functions under the integral signs are complex-valued.
In principle, with the help of powerful computers (at considerable computing time and expense) such a set can be solved for a finite and small number of equations. The corresponding algorithms are very sophisticated though, and even for the simplest non-linear networks cannot be realized in a real-time mode.
The mentioned set of integral equations for the special case of a non-linear optical network has been solved, as described in "Holographic Analysis of the Characteristics of Relief Recording Media," Opt. Spectrosc. (USSR), 58(4), Apr. 1985, pp. 533-537; and "Holographic Analysis of Phase Media with Absorption," Opt. Spectrosc. (USSR), 60(3), March 1986, pp. 365-369 (referred to hereinafter as "ref. [1]" and "ref. [2]", respectively), of which the inventor is an author, with the solutions being represented by simple algebraic equations. The inventor has now formulated a general solution applicable in the context of any continuous nonlinear network and has formulated both an experimental procedure and corresponding programmed microprocessor for the evaluation of transfer functions.